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Magazine Chapter 11: Problem 36
Sketch the surface. $$ z=\sqrt{1+x^{2}+y^{2}} $$
Short Answer
Expert verified
The surface is an elliptic paraboloid "dome-like" shape centered at z=1.
Step by step solution
01
Identify the Surface Type
The given equation is \( z = \sqrt{1 + x^2 + y^2} \). This is a scalar equation representing a surface in three dimensions. Recognize the equation as defining a paraboloid of revolution, specifically an elliptic paraboloid that opens upward.
02
Simplify the Equation for Analysis
Rewrite the original equation as \( z^2 = 1 + x^2 + y^2 \) to make it easier to understand the shape of the surface. This form is useful to see that \( z^2 - x^2 - y^2 = 1 \), which is similar to the standard form of a hyperboloid of two sheets, but due to the square root the surface takes the shape of a paraboloid.
03
Determine the Domain of the Function
The function \( z = \sqrt{1 + x^2 + y^2} \) is defined for all real \( x \) and \( y \). Since the square root function is always non-negative, the value of \( z \) will always be greater than or equal to 1.
04
Analyze the Surface Shape
The surface has a circular symmetry because inside the square root, the terms \( x^2 \) and \( y^2 \) create circles of constant radius. As \( x \) and \( y \) increase in magnitude, \( z \) increases as well. This indicates a bump or dome-like shape centered at \( z = 1 \) when both \( x = 0 \) and \( y = 0 \).
05
Sketch the Surface
To sketch the surface, start with base contours at different heights of \( z \). For example, when \( z = 1 \), \( x = 0 \) and \( y = 0 \); as \( z \) increases, radii increase describe circular levels. Draw several contours and connect them smoothly to illustrate the curved surface of the paraboloid opening upward.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Elliptic Paraboloid
An elliptic paraboloid is a three-dimensional surface that can be visualized by extending the concept of a parabola. This surface can be thought of as a parabolic shape that is revolved around a vertical axis, creating a bowl or dome-like form. In the equation \( z = \sqrt{1 + x^2 + y^2} \), we see a special case of an elliptic paraboloid. Here, the surface "opens upward," as evident from the increasing value of \( z \) with the magnitude of \( x \) and \( y \).
This elliptic paraboloid has a circular symmetry about the \( z \)-axis because both \( x^2 \) and \( y^2 \) contribute equally to the equation, forming circles of constant radius on horizontal cross-sections.
This elliptic paraboloid has a circular symmetry about the \( z \)-axis because both \( x^2 \) and \( y^2 \) contribute equally to the equation, forming circles of constant radius on horizontal cross-sections.
- A paraboloid is defined by its cross-sections which are parabolas along planes parallel to its axis.
- The elliptic nature means these parabolas are revolved around an axis, creating a symmetrical shape.
Surface Sketching
To sketch a surface like the elliptic paraboloid, start by understanding the behavior of the function in steps. By transforming the given equation \( z = \sqrt{1 + x^2 + y^2} \) into \( z^2 = 1 + x^2 + y^2 \), we can analyze the shape more intuitively.
Start by plotting the main feature: contour lines or level sets of the surface. These are horizontal slices of the surface at different \( z \) values.
Start by plotting the main feature: contour lines or level sets of the surface. These are horizontal slices of the surface at different \( z \) values.
- When \( z = 1 \), the contour is a point at \( x = 0 \), \( y = 0 \).
- For \( z > 1 \), each level set is a circle centered at the origin whose radius increases as \( z \) increases.
Three-Dimensional Geometry
Three-dimensional geometry helps us visualize the elliptic paraboloid in space. It requires stepping beyond the two-dimensional representations and conceptualizing the interaction between variables \( x \), \( y \), and \( z \).
In this context, geometric intuition involves imagining how every point \( (x, y, z) \) fits into the broader picture of a 3D shape.
In this context, geometric intuition involves imagining how every point \( (x, y, z) \) fits into the broader picture of a 3D shape.
- The origin \((x = 0, y = 0, z = 1)\) acts like a minimum point on our surface.
- For any real values of \( x \) and \( y \), the formula \( z = \sqrt{1 + x^2 + y^2} \) remains non-negative, suggesting the elliptic paraboloid rises continually upwards.
- Lines of symmetry are essential as they enforce the circular nature of the changes in \( x \) and \( y \).
